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Lifetime Random Regression Model for Test Day Milk Yields
Runqing Yang and L. R. Schaeffer
CGIL, Department of Animal & Poultry Science,
University of Guelph,
Guelph, Ontario, Canada, N1G 2W1

INTRODUCTION The current genetic evaluation system for production traits in Canada uses a random regression model (RRM). Each lactation is a different trait, but only 3 lactations are included in the analyses due to computer limitations on memory and time. Within each lactation are milk, fat, and protein yields, and somatic cell scores giving a total of 12 traits overall. Initially Wilmink's function was used as the submodel for the random regressions, and this function has 3 parameters. Thus, for 12 traits there were a total of 36 parameters to estimate for each animal's additive genetic effects. Going to Legendre polynomials will increase this to 48 or even 60 parameters per animal. To add more lactations would increase the number of parameters by 20 per lactation.

An alternative would be to use a lifetime RRM in which lactation number and days in milk (DIM) are included simultaneously in one model for each of milk, fat, protein, and somatic cell scores. By including regressions on parity number (PN) as well as on DIM, virtually any number of lactations could be included without increasing the number of parameters to be estimated per animal. The questions surrounding a lifetime RRM are as follows:

1.

Is the dairy industry interested in adding later lactation test day yields into genetic evaluations?

2.

If yes, what is the best RRM for including these later lactations?

3.

What is the added value to genetic evaluations relative to the costs of running a lifetime RRM?

4.

How to express lifetime production evaluations?

DATA

Test day records of Canadian Jerseys calving between 1988 and 1999 were used in this study. Only test day records from the first 6 parities were kept and only if they were between 5 DIM and 330 DIM (longer than the current methods). Cows had to have TD records in their first lactation to be included in the study. This was to be able to properly account for culling of cows. Below is a table of the numbers of cows, TD records, and mean milk yield and standard deviations per lactation of the records kept in the study.

Table 1
Description of Data

Parity	Cows	TDrecs	Mean	SD
1	10,948	93,629	16.3	4.6
2	7,651	61,797	19.0	6.1
3	4,953	40,547	20.3	6.6
4	2,908	23,002	20.8	6.7
5	1,602	12,234	20.9	6.6
6	810	5,876	20.8	6.7

MODEL

The basic lifetime RRM can be written as follows:

$$\begin{eqnarray*}y & = & HTDP \\ & & + f(TAPS, \ DIM, \ 5) \\ & & + r(add, \ sub ) + r(PE, \ sub) \\ & & + e(DIM) \end{eqnarray*}$$

where y is a test day milk yield on a cow in parity p, at DIM d; HTDP is the herd-test date-parity effect; $f(TAPS, \ DIM, \ 5)$ is a set of fixed regressions that model the shape of the lactation curves for cows in the Time-Age-Parity-Season subclass at DIM days in milk, and these regressions are modeled by polynomials of order 5 (in this study); $r(add, \ sub)$ are random regressions for animal additive genetic effects, and sub refers to the submodel used for the random regressions (several different submodels were examined in this study and are described later); $r(PE, \ sub)$ are random regressions for animal permanent environmental effects which could have a different submodel from that for the additive genetic effects; and e(DIM) are the residual effects assumed to have a different variance depending on the days in milk.

The submodel may be written as follows: sub & = & b₀₀ + r(DIM*PN,m*n)

where b₀₀ is an overall intercept, and r(DIM*PN,m*n) are random regressions on the interaction of DIM and PN Legendre polynomials of order m for DIM(of which m=4 for this report) and of order n for PN. Previous work did not include the interaction term and assumed that the functions on DIM were the same regardless of parity number.

Submodels that included b₀₀ resulted in heritability estimates for parity 6 that were very much higher than the estimates for the other lactations. Therefore, the intercept was removed from several analyses. The order of the Legendre polynomials for PN was increased from n=1 to 3. Table 2 shows the estimated heritabilities for 330 d yields and the residual variances by lactation number. As n increased, the residual variances decreased. The residual variance was assumed to be the same for lactations 5 and 6 because there were few records in each of the 30 DIM classes within lactations.

Table 2
Heritabilities(h²) and Residual Variances (VE)
by Parity Number and Submodel

	n=1		n=2		n=3
Parity	h2	VE	h2	VE	h2	VE
1	.33	857	.31	791	.31	771
2	.26	2013	.28	1783	.30	1514
3	.30	2239	.31	2091	.29	2098
4	.34	2401	.31	2251	.31	2070
5	.38	2519	.33	2172	.34	1925
6	.41	2519	.37	2172	.38	1925

Discussion

Comparisons of different submodels are continuing. Once a suitable model is identified, then genetic evaluations using that model and the latest TD data will be produced for comparison to the current official RRM, at least the first three lactations. A determination of the gain in accuracy and the value of having 3 more lactations included must be made. More data will result in more computer time to obtain genetic evaluations, but maybe this will not be substantial. Van Doormaal estimated that this model would result in the addition of 20% more TD records per run than the current models.

ACKNOWLEDGMENTS

Data were provided by the Canadian Dairy Network, Guelph, Ontario, Canada. The authors are grateful to the Ontario Ministry of Agriculture, DairyGen, and the Natural Science and Engineering Research Council for their financial support.

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